Another Utility Sink Install

Hey guys, I'm attempting my first DIY plumbing project.
I found this site, and have learned quite a bit, thanks so much to everyone who gives their time & advice here.
I'm going to put a utility sink in the garage next to the laundry.
Here's what I got;
I've seen the example picture on a few of the threads, I guess I just want a little confirmation, is this all I need to do;

What you have drawn in is fine as long as you install a sanitary tee above the existing one for the drain. If you are going to revent the washer, the vent take-off should be a wye and 45. The drain should be 2" and vent 1-1/2".

You need to make sure that the sink is high enough so that when the trap is installed this will allow proper pitch to the drain.

Because the washer drains a high volume of water quickly, the tee for the sink should be above the tee for the washer.

Putting the sink tee over the washer makes it necessary to install a re-vent on the washer drain to meet code in many areas.

An additional challenge is that the sink drain and the washer drain need to be 2" Because the existing tee for the washer is only 1-1/2 inch at the top, it that existing tee will need to be replaced with one that is 2" at all 3 ports.

You are going to need to cut out the existing tee and laundry drain and start fresh.

Don't cut off too much or you wont have anything to connect to, and/or the washer trap will be on the floor.

Thanks again for all the help everyone.
Here is what I've ended up with before glueing and and going through the studs.
The washer drain is going to be higher, I haven't figured out where I'm putting the outlet box yet, other that that, how am I looking.

Funny how towns ravaged by natural disaster, with massive amounts of total rehabilitation, and permits/inspections to go with it, tend to lax up a bit on code enforcement. Seems they have better things to do than insist that the letter of code be enforced, such as get millions of people back in their homes.

I'm no code-nazi. You could probably look up many posts on here where that's apparent. But in this case, there was no reason to not do it right, the access wasn't that difficult, the wall is open, etc.

I'll let the real pros like Terry answer for sure, but what you have there looks all right to me. Make sure you get your full standpipe height on that washer drain, but you already mentioned that you know you need to do that.

Isn't a horizontal to horizontal 90 generally a long sweep situation as well? I could be remembering wrong as far as code, but I'd always do a long sweep or 2 45s in that situation if possible. I know a medium 90 is usually used for a trap arm turning out of the wall (such as in this setup), but I never use a medium 90 on its side anywhere else in the drainage system.

I find that vent els can be handy in a few select situations, but yes, generally unnecessary extra parts.

Like the cleanout in your setup too, btw. Often skipped, but a nice thing to put in while you're working. Just be sure to leave some way of getting at it through the drywall, so you can find it when you need it